π x a + e 2 ) + c 3 sin ( 6 π x a + e 3 ) + etc.
In this formula , , , , etc., and also , , , etc., are constants, chosen so as to suit the particular function. Again we have to ask, How many terms have to be chosen? And here a new difficulty arises: for we can prove that, though in some particular cases a definite number will do, yet in general all we can do is to approximate as closely as we like to the value of the function by taking more and more terms. This process of gradual approximation brings us to the consideration of the theory of infinite series, an essential part of mathematical theory which we will consider in the next chapter.
The above method of expressing a periodic
function as a sum of sines is called the "harmonic analysis" of the function. For example, at any point on the sea coast the tides rise and fall periodically. Thus at a point near the Straits of Dover there will be two daily tides due to the rotation of the earth. The daily rise and fall of the tides are complicated by the fact that there are two tidal waves, one coming up the English Channel, and the other which has swept round the North of Scotland, and has then come southward down the North Sea. Again some high tides are higher than others: this is due to the fact that the Sun has also a tide-generating influence as well as the Moon. In this way monthly and other periods are introduced.
We leave out of account the exceptional influence of winds which cannot be foreseen. The general problem of the harmonic analysis of the tides is to find sets of terms like those in the expression on [page]191 above, such that each set will give with approximate accuracy the contribution of the tide-generating influences of one "period" to the height of the tide at any instant. The argument will therefore be the time reckoned from any convenient commencement.